16. Bibliography
- 1
Xbraid: parallel multigrid in time. http://llnl.gov/casc/xbraid.
- 2
D. G. Anderson. Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Machinery, 12:547–560, 1965.
- 3
U. M. Ascher and L. R. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, Pa, 1998.
- 4
Cody J Balos, David J Gardner, Carol S Woodward, and Daniel R Reynolds. Enabling GPU accelerated computing in the SUNDIALS time integration library. Parallel Computing, 108:102836, 2021.
- 5
R.E. Bank, W.M. Coughran, W. Fichtner, E.H. Grosse, D.J. Rose, and R.K. Smith. Transient simulation of silicon devices and circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 4():436–451, 1985. doi:10.1109/TCAD.1985.1270142.
- 6
S.R Billington. Type-insensitive codes for the solution of stiff and nonstiff systems of ordinary differential equations. Master Thesis, University of Manchester, United Kingdom, 1983.
- 7
David Boehme, Todd Gamblin, David Beckingsale, Peer-Timo Bremer, Alfredo Gimenez, Matthew LeGendre, Olga Pearce, and Martin Schulz. Caliper: performance introspection for hpc software stacks. In SC'16: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, 550–560. IEEE, 2016.
- 8
P. Bogacki and L.F. Shampine. A 3(2) pair of Runge–Kutta formulas. Applied Mathematics Letters, 24():407–434, 1989.
- 9
K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM, Philadelphia, Pa, 1996.
- 10
P. N. Brown. A local convergence theory for combined inexact-Newton/finite difference projection methods. SIAM J. Numer. Anal., 24(2):407–434, 1987.
- 11
P. N. Brown, G. D. Byrne, and A. C. Hindmarsh. VODE, a Variable-Coefficient ODE Solver. SIAM J. Sci. Stat. Comput., 10:1038–1051, 1989.
- 12
P. N. Brown and A. C. Hindmarsh. Reduced Storage Matrix Methods in Stiff ODE Systems. J. Appl. Math. & Comp., 31:49–91, 1989.
- 13
P. N. Brown, A. C. Hindmarsh, and L. R. Petzold. Using Krylov Methods in the Solution of Large-Scale Differential-Algebraic Systems. SIAM J. Sci. Comput., 15:1467–1488, 1994.
- 14
P. N. Brown, A. C. Hindmarsh, and L. R. Petzold. Consistent Initial Condition Calculation for Differential-Algebraic Systems. SIAM J. Sci. Comput., 19:1495–1512, 1998.
- 15
P. N. Brown and Y. Saad. Hybrid Krylov Methods for Nonlinear Systems of Equations. SIAM J. Sci. Stat. Comput., 11:450–481, 1990.
- 16
J.C. Butcher. Numerical Methods for Ordinary Differential Equations. Wiley, Chicester, England, 2 edition, 2008.
- 17
G. D. Byrne. Pragmatic Experiments with Krylov Methods in the Stiff ODE Setting. In J.R. Cash and I. Gladwell, editors, Computational Ordinary Differential Equations, 323–356. Oxford, 1992. Oxford University Press.
- 18
G. D. Byrne and A. C. Hindmarsh. A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations. ACM Trans. Math. Softw., 1:71–96, 1975.
- 19
G. D. Byrne and A. C. Hindmarsh. User Documentation for PVODE, An ODE Solver for Parallel Computers. Technical Report UCRL-ID-130884, LLNL, May 1998.
- 20
G. D. Byrne and A. C. Hindmarsh. PVODE, An ODE Solver for Parallel Computers. Intl. J. High Perf. Comput. Apps., 13(4):254–365, 1999.
- 21
Y. Cao, S. Li, L. R. Petzold, and R. Serban. Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and its Numerical Solution. SIAM J. Sci. Comput., 24(3):1076–1089, 2003.
- 22
M. Caracotsios and W. E. Stewart. Sensitivity Analysis of Initial Value Problems with Mixed ODEs and Algebraic Equations. Computers and Chemical Engineering, 9:359–365, 1985.
- 23
J.R. Cash. Diagonally implicit runge-kutta formulae with error estimates. IMA Journal of Applied Mathematics, 24():293–301, 1979.
- 24
J.R. Cash and A.H. Karp. A variable order runge-kutta method for initial value problems with rapidly varying right-hand sides. ACM Transactions on Mathematical Software, 16():201–222, 1990.
- 25
Rujeko Chinomona and Daniel R Reynolds. Implicit-explicit multirate infinitesimal gark methods. SIAM Journal on Scientific Computing, 43(5):A3082–A3113, 2021.
- 26
S. D. Cohen and A. C. Hindmarsh. CVODE User Guide. Technical Report UCRL-MA-118618, LLNL, September 1994.
- 27
S. D. Cohen and A. C. Hindmarsh. \mbox CVODE, a Stiff/Nonstiff ODE Solver in C. Computers in Physics, 10(2):138–143, 1996.
- 28
A. M. Collier, A. C. Hindmarsh, R. Serban, and C.S. Woodward. User Documentation for KINSOL v6.2.0. Technical Report UCRL-SM-208116, LLNL, 2022.
- 29
A. M. Collier and R. Serban. Example Programs for KINSOL v6.2.0. Technical Report UCRL-SM-208114, LLNL, 2022.
- 30
T. A. Davis and P. N. Ekanathan. Algorithm 907: KLU, a direct sparse solver for circuit simulation problems. ACM Trans. Math. Softw., 2010.
- 31
R. S. Dembo, S. C. Eisenstat, and T. Steihaug. Inexact Newton Methods. SIAM J. Numer. Anal., 19:400–408, 1982.
- 32
J. W. Demmel, J. R. Gilbert, and X. S. Li. An asynchronous parallel supernodal algorithm for sparse gaussian elimination. SIAM J. Matrix Analysis and Applications, 20(4):915–952, 1999.
- 33
J. E. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996.
- 34
J.R. Dormand and P.J. Prince. A family of embedded runge-kutta formulae. Journal of Computational and Applied Mathematics, 6():19–26, 1980.
- 35
M.R. Dorr, J.-L. Fattebert, M.E. Wickett, J.F. Belak, and P.E.A. Turchi. A numerical algorithm for the solution of a phase-field model of polycrystalline materials. Journal of Computational Physics, 229(3):626–641, 2010.
- 36
Edda Eich. Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM Journal on Numerical Analysis, 30(5):1467–1482, 1993.
- 37
S. C. Eisenstat and H. F. Walker. Choosing the Forcing Terms in an Inexact Newton Method. SIAM J. Sci. Comput., 17:16–32, 1996.
- 38
R.D. Falgout, S. Friedhoff, TZ.V. Kolev, S.P. MacLachlan, and J.B. Schroder. Parallel time integration with multigrid. SIAM Journal of Scientific Computing, 36():C635–C661, 2014.
- 39
H. Fang and Y. Saad. Two classes of secant methods for nonlinear acceleration. Numer. Linear Algebra Appl., 16:197–221, 2009.
- 40
W. F. Feehery, J. E. Tolsma, and P. I. Barton. Efficient Sensitivity Analysis of Large-Scale Differential-Algebraic Systems. Applied Numer. Math., 25(1):41–54, 1997.
- 41
E. Fehlberg. Low-order classical runge-kutta formulas with step size control and their application to some heat transfer problems. Technical Report 315, NASA, 1969.
- 42
R. W. Freund. A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems. SIAM J. Sci. Comp., 14:470–482, 1993.
- 43
Laura Grigori, James W. Demmel, and Xiaoye S. Li. Parallel symbolic factorization for sparse LU with static pivoting. SIAM J. Scientific Computing, 29(3):1289–1314, 2007.
- 44
K. Gustafsson. Control theoretic techniques for stepsize selection in explicit runge-kutta methods. ACM Transactions on Mathematical Software, 17():533–554, 1991.
- 45
K. Gustafsson. Control theoretic techniques for stepsize selection in implicit runge-kutta methods. ACM Transactions on Mathematical Software, 20():496–512, 1994.
- 46
E. Hairer, S. P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I. Springer-Verlag, Berlin, 1987.
- 47
E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1991.
- 48
Vicente Hernández, José E Román, and Andrés Tomás. A parallel variant of the gram-schmidt process with reorthogonalization. In PARCO, 221–228. 2005.
- 49
M. R. Hestenes and E. Stiefel. Methods of Conjugate Gradients for Solving Linear Systems. J. Research of the National Bureau of Standards, 49(6):409–436, 1952.
- 50
K. L. Hiebert and L. F. Shampine. Implicitly Defined Output Points for Solutions of ODEs. Technical Report SAND80-0180, Sandia National Laboratories, February 1980.
- 51
A. C. Hindmarsh. Detecting Stability Barriers in BDF Solvers. In J.R. Cash and I. Gladwell, editor, Computational Ordinary Differential Equations, 87–96. Oxford, 1992. Oxford University Press.
- 52
A. C. Hindmarsh. Avoiding BDF Stability Barriers in the MOL Solution of Advection-Dominated Problems. Appl. Num. Math., 17:311–318, 1995.
- 53
A. C. Hindmarsh. The PVODE and IDA Algorithms. Technical Report UCRL-ID-141558, LLNL, December 2000.
- 54
A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward. \mbox SUNDIALS, suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw., pages 363–396, 2005.
- 55
A. C. Hindmarsh and R. Serban. User Documentation for CVODE v6.2.0. Technical Report UCRL-SM-208108, LLNL, 2022.
- 56
A. C. Hindmarsh, R. Serban, and A. Collier. Example Programs for IDA v6.2.0. Technical Report UCRL-SM-208113, LLNL, 2022.
- 57
A. C. Hindmarsh, R. Serban, and D. R. Reynolds. Example Programs for CVODE v6.2.0. Technical Report, LLNL, 2022. UCRL-SM-208110.
- 58
A. C. Hindmarsh and A. G. Taylor. PVODE and KINSOL: Parallel Software for Differential and Nonlinear Systems. Technical Report UCRL-ID-129739, LLNL, February 1998.
- 59
K. R. Jackson and R. Sacks-Davis. An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs. ACM Trans. Math. Softw., 6:295–318, 1980.
- 60
Seth R. Johnson, Andrey Prokopenko, and Katherine J. Evans. Automated fortran-c++ bindings for large-scale scientific applications. 2019. URL: http://arxiv.org/abs/1904.02546, arXiv:1904.02546.
- 61
C. T. Kelley. Iterative Methods for Solving Linear and Nonlinear Equations. SIAM, Philadelphia, 1995.
- 62
C.A. Kennedy and M.H. Carpenter. Additive runge-kutta schemes for convection-diffusion-reaction equations. Applied Numerical Mathematics, 44():139–181, 2003.
- 63
C.A. Kennedy and M.H. Carpenter. Higher-order additive runge–kutta schemes for ordinary differential equations. Applied Numerical Mathematics, 136():183–205, 2019.
- 64
O. Knoth and R. Wolke. Implicit-explicit runge–kutta methods for computiong atmospheric reactive flows. Applied Numerical Analysis, 28(2):327–341, 1998.
- 65
A. Kværno. Singly diagonally implicit runge-kutta methods with an explicit first stage. BIT Numerical Mathematics, 44():489–502, 2004.
- 66
S. Li, L. R. Petzold, and W. Zhu. Sensitivity Analysis of Differential-Algebraic Equations: A Comparison of Methods on a Special Problem. Applied Num. Math., 32:161–174, 2000.
- 67
X. S. Li. An overview of SuperLU: algorithms, implementation, and user interface. ACM Trans. Math. Softw., 31(3):302–325, September 2005.
- 68
X.S. Li, J.W. Demmel, J.R. Gilbert, L. Grigori, M. Shao, and I. Yamazaki. SuperLU Users' Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory, September 1999. http://crd.lbl.gov/ xiaoye/SuperLU/. Last update: August 2011.
- 69
Xiaoye S. Li and James W. Demmel. SuperLU_DIST: a scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans. Mathematical Software, 29(2):110–140, June 2003.
- 70
P. A. Lott, H. F. Walker, C. S. Woodward, and U. M. Yang. An accelerated Picard method for nonlinear systems related to variably saturated flow. Adv. Wat. Resour., 38:92–101, 2012.
- 71
T. Maly and L. R. Petzold. Numerical Methods and Software for Sensitivity Analysis of Differential-Algebraic Systems. Applied Numerical Mathematics, 20:57–79, 1997.
- 72
J. M. Ortega and W. C. Rheinbolt. Iterative solution of nonlinear equations in several variables. SIAM, Philadelphia, 2000. Originally published in 1970 by Academic Press.
- 73
D.B. Ozyurt and P.I. Barton. Cheap second order directional derivatives of stiff ODE embedded functionals. SIAM J. of Sci. Comp., 26(5):1725–1743, 2005.
- 74
K. Radhakrishnan and A. C. Hindmarsh. Description and Use of LSODE, the Livermore Solver for Ordinary Differential Equations. Technical Report UCRL-ID-113855, LLNL, march 1994.
- 75
Daniel R. Reynolds. Example Programs for ARKODE v5.2.0. Technical Report, Southern Methodist University, 2022.
- 76
Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput., 14(2):461–469, 1993. doi:http://dx.doi.org/10.1137/0914028.
- 77
Y. Saad and M. H. Schultz. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comp., 7:856–869, 1986.
- 78
A. Sandu. A class of multirate infinitesimal gark methods. SIAM Journal of Numerical Analysis, 57():2300–2327, 2019.
- 79
A. Sayfy and A. Aburub. Embedded additive runge-kutta methods. International Journal of Computer Mathematics, 79():945–953, 2002.
- 80
M. Schlegel, O. Knoth, M. Arnold, and R. Wolke. Multirate runge–kutta schemes for advection equations. Journal of Computational Applied Mathematics, 226():345–357, 2009.
- 81
M. Schlegel, O. Knoth, M. Arnold, and R. Wolke. Implementation of multirate time integration methods for air pollution modelling. GMD, 5():1395–1405, 2012.
- 82
M. Schlegel, O. Knoth, M. Arnold, and R. Wolke. Numerical solution of multiscale problems in atmospheric modeling. Applied Numerical Mathematics, 62():1531–1542, 2012.
- 83
R. Serban and A. C. Hindmarsh. \mbox CVODES, the sensitivity-enabled ODE solver in \mbox SUNDIALS. In Proceedings of the 5th International Conference on Multibody Systems, Nonlinear Dynamics and Control. Long Beach, CA, 2005. ASME.
- 84
R. Serban and A. C. Hindmarsh. Example Programs for CVODES v6.2.0. Technical Report UCRL-SM-208115, LLNL, 2022.
- 85
L. F. Shampine. Implementation of implicit formulas for the solution of ODEs. SIAM Journal on Scientific and Statistical Computing, 1(1):103–118, 1980.
- 86
LF Shampine. Conservation laws and the numerical solution of odes, ii. Computers & Mathematics with Applications, 38(2):61–72, 1999.
- 87
G. Soderlind. The automatic control of numerical integration. CWI Quarterly, 11():55–74, 1998.
- 88
G. Soderlind. Digital filters in adaptive time-stepping. ACM Transactions on Mathematical Software, 29():1–26, 2003.
- 89
G. Soderlind. Time-step selection algorithms: adaptivity, control and signal processing. Applied Numerical Mathematics, 56():488–502, 2006.
- 90
Kasia Swirydowicz, Julien Langou, Shreyas Ananthan, Ulrike Yang, and Stephen Thomas. Low synchronization gram-schmidt and gmres algorithms. Numerical Linear Algebra with Applications, 02 2020.
- 91
Stanimire Tomov, Jack Dongarra, and Marc Baboulin. Towards dense linear algebra for hybrid GPU accelerated manycore systems. Parallel Computing, 36(5-6):232–240, June 2010. doi:10.1016/j.parco.2009.12.005.
- 92
J.H Verner. Explicit runge-kutta methods with estimates of the local truncation error. SIAM Journal of Numerical Analysis, 15():772–790, 1978.
- 93
H. A. Van Der Vorst. Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comp., 13:631–644, 1992.
- 94
H. F. Walker and P. Ni. Anderson acceleration for fixed-point iterations. SIAM Jour. Num. Anal., 49(4):1715–1735, 2011.
- 95
J.A. Zonneveld. Automatic integration of ordinary differential equations. Technical Report R743, Mathematisch Centrum, Postbus 4079, 1009AB Amsterdam, 1963.
- 96
N.a. AMD ROCm Documentation. https://rocmdocs.amd.com/en/latest/index.html.
- 97
N.a. Intel oneAPI Programming Guide. https://software.intel.com/content/www/us/en/develop/documentation/oneapi-programming-guide/top.html.
- 98
N.a. KLU Sparse Matrix Factorization Library. http://faculty.cse.tamu.edu/davis/suitesparse.html.
- 99
N.a. NVIDIA CUDA Programming Guide. https://docs.nvidia.com/cuda/index.html.
- 100
N.a. NVIDIA cuSOLVER Programming Guide. https://docs.nvidia.com/cuda/cusolver/index.html.
- 101
N.a. NVIDIA cuSPARSE Programming Guide. https://docs.nvidia.com/cuda/cusparse/index.html.
- 102
N.a. SuperLU_DIST Parallel Sparse Matrix Factorization Library. http://crd-legacy.lbl.gov/ xiaoye/SuperLU/.
- 103
N.a. SuperLU_MT Threaded Sparse Matrix Factorization Library. http://crd-legacy.lbl.gov/ xiaoye/SuperLU/.